A262000 a(n) = n^2*(7*n - 5)/2.

0, 1, 18, 72, 184, 375, 666, 1078, 1632, 2349, 3250, 4356, 5688, 7267, 9114, 11250, 13696, 16473, 19602, 23104, 27000, 31311, 36058, 41262, 46944, 53125, 59826, 67068, 74872, 83259, 92250, 101866, 112128, 123057, 134674, 147000, 160056, 173863, 188442, 203814, 220000

Offset: 0

Author

Bruno Berselli, Sep 08 2015

Keywords

• nonn
• easy

Comments

Also, structured enneagonal prism numbers.

Examples

			For n=8, a(8) = 8*(7*0+1)+8*(7*1+1)+8*(7*2+1)+8*(7*3+1)+8*(7*4+1)+8*(7*5+1)+8*(7*6+1)+8*(7*7+1) = 1632.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. similar sequences with the formula n^2*(k*n - k + 2)/2: A000290 (k=0), A002411 (k=1), A000578 (k=2), A050509 (k=3), A015237 (k=4), A006597 (k=5), A100176 (k=6), this sequence (k=7), A103532 (k=8).

Cf. A001106, A069125.

Programs

Magma

[n^2*(7*n-5)/2: n in [0..40]];

Mathematica

Table[n^2 (7 n - 5)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,18,72},50] (* Harvey P. Dale, Oct 04 2016 *)

PARI

vector(40, n, n--; n^2*(7*n-5)/2)

Sage

[n^2*(7*n-5)/2 for n in (0..40)]

Formula

G.f.: x*(1 + 14*x + 6*x^2)/(1 - x)^4.

a(n) = Sum_{i=0..n-1} n*(7*i+1) for n>0, a(0)=0.

a(n+1) + a(-n) = A069125(n+1).

Sum_{i>0} 1/a(i) = 1.082675669875907610300284768825... = (42*(log(14) + 2*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14))) + 21*Pi*tan(3*Pi/14))/75 - Pi^2/15. - Vaclav Kotesovec, Oct 04 2016

From Elmo R. Oliveira, Aug 06 2025: (Start)

E.g.f.: exp(x)*x*(2 + 16*x + 7*x^2)/2.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)